If the question is unclear, off topic or has other problems, please explain.
Consider the following sets that are infinitely countable and dense in $\mathbb{R}$
$T_1=\left\{\left.\frac{M_1(n_1)}{R_1(q_1)}\right|n_1,q_1\in\mathbb{Z}\right\}$ $,T_2=\left\{\left.\frac{M_2(n_2)}{R_2(q_2)}\right|n_2,q_2\in\mathbb{Z}\right\},...,$ $T_m=\left\{\left.\frac{M_m(n_m)}{R_m(q_m)}\right|n_m,q_m\in\mathbb{Z}\right\}$
Where $M_1(x),...,M_m(x)$ and $R_1(x),...,R_m(x)$ are unique functions. For example, $M_1(x)=2x+1$, $R_1(x)=2^x$, $M_2(x)=5x+1$, $R_2(x)=\ln(x)$ and so forth.
I want to construct a measure that compares the size of each set to the size of all the sets.
"It is indeed possible to get measures that behave in a nontrivial fashion on countable subsets, and indeed the axiom of choice or some weaker forms thereof are relevant here. Namely, one can get a finitely-additive (not $\sigma$-additive) measure $\xi$ on the set of subsets of $\mathbb{N}$ such that $\xi(S)=0$ for any finite set $S\subseteq\mathbb{N}$ and $\xi(S)=1$ for any cofinite subset (i.e., subset with a finite complement). Here $\xi$ takes only two values, zero or one. For sets that are neither finite nor cofinite, the measure behaves in a nontrivial way but always exactly one of a pair of complementary sets has measure one. Such measures are used in ultraproduct constructions."
I read articles on ultrafilters and ultraproducts, but the mathematics is far beyond my knowledge.
How does one assign measures, that use ultraproduct constructions, to sets $T_1,T_2,..,T_m$?